JEE Mains · Maths · STD 12 - 8. Application and integration
If the area of the region {(x, y) : \( 1-2x\le y\le4-x^{2}, x\ge0,y\ge0 \)} is \( \frac{\alpha}{\beta}, \alpha, \beta \in N \), gcd(\(α,β\))=1, then the value of \( (\alpha+\beta) \) is :
- A 73
- B 85
- C 91
- D 67
Answer & Solution
Correct Answer
(A) 73
Step-by-step Solution
Detailed explanation
Required area \( =\frac{2}{3}\times8-\frac{1}{2}\times\frac{1}{2}\times1 \) \( =\frac{16}{3}-\frac{1}{4}=\frac{61}{12}=\frac{\alpha}{\beta} \) \( \Rightarrow\alpha+\beta=73 \)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- A straight line \(L\) through the point \((3, - 2)\) is inclined at an angle of \(60^o\) to the line \(\sqrt 3 x + y = 1\) . If \(L\) also intersects the \(x-\) axis, then the equation of \(L\) isJEE Mains 2015 Hard
- If \({ }^{20} \mathrm{C}_{\mathrm{r}}\) is the co-efficient of \(\mathrm{x}^{\mathrm{r}}\) in the expansion of \((1+x)^{20}\), then the value of \(\sum_{r=0}^{20} r^{2}\,\,{ }^{20} C_{r}\) is equal to :JEE Mains 2021 Hard
- If a circle \(C,\) whose radius is \(3,\) touches externally the circle, \(x^2 + y^2 + 2x - 4y - 4 = 0\) at the point \((2, 2),\) then the length of the intercept cut by circle \(c,\) on the \(x-\) axis is equal toJEE Mains 2018 Hard
- Let \(f\) be a differential function such that \(f'\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x},\left( {x > 0} \right)\) and \(f(1) \ne 4\). Then \(\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {\frac{1}{x}} \right)\)JEE Mains 2019 Hard
- Let the parabola \(y=x^2+\mathrm{p} x-3\), meet the coordinate axes at the points \(\mathrm{P}, \mathrm{Q}\) and R . If the circle C with centre at \((-1,-1)\) passes through the points \(P, Q\) and \(R\), then the area of \(\triangle P Q R\) is :JEE Mains 2025 Medium
- If \(\sin \left(\frac{y}{x}\right)=\log _0|x|+\frac{\alpha}{2}\) is the solution of the differential equation \(x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x\) and \(y(1)=\frac{\pi}{3}\), then \(\alpha^2\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- Let \(A=\{-2,-1,0,1,2,3\}\). let R be a relation on A defined by \(x R y\) if and only if \(y=\max \{x, 1\}\). Let \(l\) be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then \(l+\mathrm{m}+\mathrm{n}\) is equal toJEE Mains 2025 Easy
- If \(a_r\) is the coefficient of \(x^{10-r}\) in the Binomial expansion of \((1+x)^{10}\), then \(\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2\) is equal toJEE Mains 2023 Hard
- The value of \(\int_{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y\) is :JEE Mains 2024 Hard
- A plane passing through the point \((3,1,1)\) contains two lines whose direction ratios are \(1 ,-2,2\) and \(2,3,-1\) respectively. If this plane also passes through the point \((\alpha,-3,5),\) then \(\alpha\) is equal toJEE Mains 2020 Hard
- If the domain of the function
\(f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}\) is \((a, b)\), then \((1+a)^2+b^2\) is equal to :JEE Mains 2025 Easy - Let the set of all positive values of \(\lambda\), for which the point of local minimum of the function \(\left(1+x\left(\lambda^2-x^2\right)\right)\) satisfies \(\frac{x^2+x+2}{x^2+5 x+6}<0\), be \((\alpha, \beta)\). Then \(\alpha^2+\beta^2\) is equal to ...........JEE Mains 2024 Hard